Archimedean maps of higher genera

Karabáš, Ján; Nedela, Roman

doi:10.1090/s0025-5718-2011-02502-0

Cited by 17 publications

(168 citation statements)

References 20 publications

Supporting

Mentioning

168

Contrasting

Order By: Relevance

“…A dictionary that would translate between the two notations would certainly be useful. The reader is referred to [26][27][28][29][30] for related work and for further references. In particular, this theory can be extended to hypermaps [28,31].…”

Section: Discussionmentioning

confidence: 99%

Operations on Oriented Maps

2017

View full text Add to dashboard Cite

A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs.Keywords: map; oriented map; truncation; dual; medial; snub; flag graph; arc graph MSC: 52C20, 05C10, 51M20 Maps and Oriented MapsA map M on a closed surface is a two-cell embedding of a finite connected graph G = (V, E) (see, e.g., [1] or [2]). Equivalently, every map can be viewed as a set of three fixed-point-free involutions r 0 , r 1 and r 2 acting on a set of flags Ω with the property that r 0 r 2 is also a fixed-point-free involution, in which case we denote the map as M = (Ω, r 0 , r 1 , r 2 ); see, for instance, [3] (p. 415). With this second point of view, each map can be described completely using a three-edge colored cubic graph, called the flag graph, whose vertices are elements of Ω, where ω 1 and ω 2 are connected by an edge colored i in the flag graph if and only if r i (ω 1 ) = ω 2 . Generally, only connected flag graphs are considered. The graph G is called the skeleton or the underlying graph of a map M.In some cases, one may relax the conditions and allow fixed points in some of the involutions r 0 , r 1 and r 2 . Such a structure describes a map in a surface with a boundary and with its flag graph containing semi-edges. We call such a map and respective flag graph degenerate.If the surface in which the map resides is orientable, then M is said to be an orientable map. There is a well-known test involving flag graphs to determine whether a given map is orientable: Proposition 1. A map is orientable if and only if its flag graph is bipartite.In a bipartite flag graph, the flags (i.e., vertices) of the map come in two color classes; the vertices in a single color class of flags are called arcs. Restricting attention to one set of arcs corresponds to choosing an orientation of the orientable map, and we call the restricted graph an oriented map; each orientable map gives rise to two oppositely oriented oriented maps. We note that in a map there are four flags per edge of the skeleton, while in an oriented map there are two arcs per edge. That is,

show abstract

Section: Discussionmentioning

confidence: 99%

Operations on Oriented Maps

2017

View full text Add to dashboard Cite

show abstract

“…In this section we show that there are only finitely many vertex-transitive polyhedra in the genus range g 2 (see Theorem 4.1 below). Note that this result concerns a whole range of genera, not any specific values of g. In fact, it is known that for each g with g 2 there are only finitely many vertex-transitive (polyhedral) maps on an orientable surface of genus g, while there are infinitely many such maps of genus g = 0 or 1, respectively (see [17,32]). Thus, for each g 2, there clearly can only be finitely many (isomorphism types of) vertex-transitive polyhedra of genus g; however, this does not directly translate into a corresponding statement for the entire genus range.…”

Section: Vertex-transitive Polyhedramentioning

confidence: 92%

The regular Grünbaum polyhedron of genus 5

Gévay

Schulte

Wills

2014

Advances in Geometry

View full text Add to dashboard Cite

We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary space E 3 . This polyhedron was originally discovered by Grünbaum in 1999, but was recently rediscovered by Brehm and Wills. We establish isomorphism of the Grünbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grünbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g 2.

show abstract

“…The vertex-transitive polyhedral maps on orientable surfaces of genus 2, 3 and 4 have been studied in [16]. The question of the existence of polyhedral maps of all the possible types on the surface with χ = −1 has been settled except for the type [5 3 , 3] (in a private communication with Dipendu Maity, will appear elsewhere).…”

Section: Polyhedral Maps Of the Type [5 3 3]mentioning

confidence: 99%

Quasi-vertex-transitive maps on the plane

Maiti

2020

Discrete Mathematics

View full text Add to dashboard Cite

Quasi-transitive maps are the homogeneous maps on the plane with finite orbits under the action of their automorphism groups. We show that there exist quasi-transitive maps of the types [p 3 , 3] for p odd and p ≥ 5, but there doesn't exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of the type [5 3 , 3].Theorem 1.1. There exist quasi-transitive maps on the plane of the types [p 3 , 3], p odd, p ≥ 5.

show abstract

Archimedean maps of higher genera

Cited by 17 publications

References 20 publications

Operations on Oriented Maps

Operations on Oriented Maps

The regular Grünbaum polyhedron of genus 5

Quasi-vertex-transitive maps on the plane

Contact Info

Product

Resources

About